As far as I know, the sealant between my fiberglass tub and vinyl floor is still doing the job, but after twenty-five years it's dingy, dirty, ugly, and uneven. I'd love to remove that old stuff and prettify the tub-floor line with new stuff. Would I be able to generate enough heat with a hair-dryer to remove the original caulking (renting a real heat gun is not an option where I live)?

For what it's worth, I poked the caulking with a brad and discovered it's still pliant below the skin. Had the tub not settled, there would not have been a problem!

Welcome to the Community Packy:
Knives and icepicks are good for removing caulk. I would not heat it up for fear of damage to the fiberglass. Try to start at one end and you can usually pull it out like a rope.
Replace the caulk with white silicone (if your tub is white) or use clear silicone for any other color.
Glenn